The US Dollar and the Colombian Peso currency exchange rate (Tasa Representativa del Mercado, TRM) is a financial series characterized by periods of high volatility. In this paper it was applied three classes of time series: the ARMA, ARMA-GARCH and Markov Switching (MS) models to represent the logreturns of the TRM between 2013-01-03 and 2020-07-02. The best models among several fitted models for each class were determined based on the Akaike and the Bayesian information criteria (AIC and BIC, respectively) and one-step forecasts in the period 2020-07-03 to 2020-07-31. The ARMA-GARCH model allowed a more precise description of the conditional variance than the ARMA model. Furthermore, the MS model defined 3 regimes each one with its own AR process. The regime with highest variability showed sporadic occurrences, and can be associated to three important events at global scale: the oil crisis in 2014, the US-China trade war in 2018, and the COVID-19 pandemics in march 2020. The results demonstrate the robustness of the models for forecasting in one-step or longer time windows.
and Random walk models, finding that the MS model performed better in a forecast horizon of 25 days and ARIMA in one-step forecasting [14].
Modeling currencies exchange rates from other Latin-american countries has also been of interest in recent years. Ortiz et al. fitted the US Dollar and Mexican Peso (MXN-USD) currency exchange rate with two classes of GARCH models with diferent distributions of the residuals and compared their performances [12]. Espinosa Gonzáles et al. implemented a MS to describe the exchange rate of the Peruvian Real (PEN-USD) finding that the best description of the series was given by a three regime model [6]. Later, Rodríguez et al. characterized the volatility of daily stock markets returns in Argentina, Brazil, Chile, Mexico and Peru, by applying extended linear models contrasting with traditional time series models [13]. Meneses and Alvarado implemented back propagation artificial neural networks to describe and predict in one-step and longer time windows the MXN-USD exchange rate [15].
However, there is a need to implement more robust models for the TRM that provide better forecasts for longer time windows. This article applies ARMA, GARCH, and MS models for the daily value of the TRM by fitting in a longer time window than the preceding works for the TRM, specifically from January 3, 2013 to July 2, 2020. This period is characterized by several events of high volatility and abrupt changes in the mean level.
In order to apply the time series models explained in this section, the data series must have a constant zero mean. Therefore, in the following it was assumed that is the log-returns of the TRM series: conditions.
= ( ) − ( −1).
Section 2 shows the statistical tests that assure that satisfies these minimal stationary 2.1. Autoregressive moving average models moving average process
( ) [1]: (, ) is constituted by the sum of an autoregressive process ( ) and a = 0 + ∑ − + ∑ − + , variance are constant in time.
where is a white noise process at time with zero mean and variance 2, the AR components are the contribution of the previous values of in the previous times, and the MA components are the contribution of the white noise until periods in the past. Under certain conditions of the parameters { } and { }, ARMA processes are stationary, i.e., their mean and where is a white noise process with zero mean and variance equal to one, and { }, { }, are parameters that need to be estimated from the data. For stability, stationarity and positive values in the series, it is necessary that , ≥ 0, ∑ =1 + ∑ =1 < 1, and > 0 [2, 3, 4].
Many financial series show abrupt changes caused by macroeconomic factors, government policies, or other circumstances [8]. This is materialized in the series through the appearance of sudden changes in their usual behavior, generating breaks in the mean and variance of the process. For instance, a relative long period of low volatility can abruptly change to a high volatility one, which in some cases can last for long periods, generating clusters in the series. This clearly implies a problem when trying to apply the assumptions of stationary.
One of the approaches is to assume that the time series parameters depend on an external environment that evolves following a markovian process, for example a discrete Markov chain. which depends only on the present state (the markovian property): In this context, the state of the Markov chain is called the regime, takes values in a countable set ( = 1, 2, … , ), and the transition between regimes and in one-step has probability
One of the main problems in time series modeling is that the conditional variance 2| −1 series upon its past values is not stationary in time (i.e., the series is heterocedastic). By defining of the the white noise with a conditional variance as an (, ), it is obtained a GARCH model. =1
A Markov Switching ( ( ) ( ), has the form: = ( +1 = | = ) = ( +1 = | = , −1 = 1, …) .
) model for with regimes and ( ) processes, denoted as = 0, + ∑ , − + . the Markov chain must be determined.
That is, the AR process followed by that depends on the regime at time has parameters , [5]. When fitting an and white noise with variance 2 ( ) ( ) to a time series, the AR parameters , for each regime = 1, … , and the one-step transition probabilities of
This section presents a descriptive analysis of the series of the TRM and its log-returns, a
methodology for the fitting and selection of the models, and the results of the fitted
ARMA,
GARCH, and MS models to the log-return series.
(
It is clear from Figure 1(a) that the TRM series is not stationary. In fact, the Dickey-Fuller test
suggests that the hypothesis of an stationary series must be rejected with a p-value of 0.9093.
With the log-return series defined by eq. (
It is important to note in Figure 1(a) that the mean of the TRM series has at least two periods of regular increase, from 2015-01 to 2016-01 and from 2018-06 to 2020-07. Besides, Figure 1(b) shows clusters of volatility in the period 2015-01 to 2016-01, from 2018-01 to 2018-06 and the period between 2020-03 an 2020-07. It is well known that three exogenous events influenced these abrupt changes in the TRM structure: the oil crisis 2014-10 that came with a quick fall of the hydrocarbons prices [16], the US-China trade war 2018-07 with duties imposed for both sides, and the global expansion of COVID-19 in 2020 [18].
The autocorrelograms of in Figure 2 suggest that the series could be an (
Model
ARMA(
BIC
AIC
It was used the R software for the fitting, selection and forecast of the time series models. An iterative methodology was implemented in order to select the best ARMA, ARMA-GARCH and MS-AR models. First, several number of models for each class were adjusted via maximumlikelihood estimation and the BIC and AIC indexes were calculated. Then, with the best five models according to the lowest values of BIC and AIC, one-step forecasts were generated at a horizon of 21 days. Finally, the errors MAPE, MAE and RMSE were calculated and the model with the lowest MAPE is chosen. This methodology assures the simplicity and precision of the selected model.
Following the previous methodology, 900 ARMA(p,q) models (with diferent orders , ) were
evaluated. The five models with the lowest and values are presented in Table 1. The
model with the best forecast is (
Finally, the one-step forecast was performed with the function forecast in R. The projected values of the best five ARMA-GARCH models for . values show a fairly good fit that hardly departs from the actual value of the series in most periods. It is notable that in times of sudden changes, the level and trend correction took few steps to perform (Figure 4).
Suppose an ARMA(, ) model for . If the white noise process follows the conditions for a GARCH ( ′, ′) process (section 2.2), then follows an ARMA(, )GARCH ( ′, ′) model.
Around 3700 combinations of ARMA-GARCH models were evaluated by varying the orders
, ,
with the best performance was ARMA(
With respect to the one-step forecast of the selected model, the projected value shows a fairly
good approximation that hardly departs from the actual value of the series. The confidence
intervals, calculated based on the one-step predicted conditional variance from the GARCH(
ARIMA model of section 3.3. Hence, the ARMA(
( ) ( ) models were adjusted using the MSwM package. The five models with the lowest
and are shown in Table 5 from 63 models evaluated. The (
Table 6 presents the values of the intercepts 0, and the standard deviation of the white noise for each regime . Note that regime 2 has the higher volatility of the three regimes, while regime 3 has the lowest by around half of the regime 1. Also, the coeficient 0 for regime 2 is the highest (in absolute terms) among the other regimes. This suggests that this regime predominates in the periods with rupture in the mean and with higher variance of the TRM. Moreover, Table 6 also shows the transition probability matrix of the Markov chain of the regimes. It is possible to observe that once in any regime = 1, 2, 3, will stay there for around 98% of the times. However, the probability of transition from regime 1 to regime 2 is very low (0.011) and almost zero from regime 3 to regime 2. This reflects the fact that periods of high volatility of the TRM series are rare but can persist for long periods. And on the other hand, there exists a major probability of transitions between regime 1 and regime 3, and vice versa. This represents the normal dynamic of the TRM with long periods of low-mid variability through time.
The smoothed probabilities show excellent segmentation of the periods where the log-returns of the TRM series show marked fluctuations (Figure 7). It is observed that regime 2 periods start approximately at the same time when three critical worldwide situations initiated: the oil crisis, starting at 2015-01, the US-China trade war starting at 2018-06 and the COVID-19 outbreak, which reached Colombia in 2020-03. These periods showed the fastest depreciation and volatility of the TRM in the last decade.
The one-step forecast of the selected model in Figure 8 shows acceptable results in the description of the average response, however the confidence intervals are wide. Figure 8 also shows the smoothed probabilities of each regime in the period 2020-05 to 2020-07. It is clear that the model develops a gradual transition from regime 2 (of high volatility) to regime 3 (of low-mid volatility), and that regime 1 loses preponderance. Regime 3 probability increases to almost 95% after an eight-day forecast. Finally, the residuals obtained from the model fit show no significant correlations after the first lag, which is verifiable from Figure 9.
When the results of the best ARMA, ARMA-GARCH, and MS models are compared, it was found that the Markov Switching model achieves the best one-step adjustment, getting lower error indicators (4% less) and index as shown in Table 7. That is, by making the mean and variance depending on diferent regimes, it was possible a better forecast of the TRM series in the short term.
When evaluated in a longer time window, in this case a horizon of 21 days, the forecast obtained with the ARMA model shows a better performance, although similar to the − model (see Table 7). Then, for this situation an ARMA model is a simpler and a more easy-to-implement representation, but the − model provides a more accurate representation of the variance.
In the case of the modeling of the series over long periods of time, the fact of being able to characterize several regimes to represent specific realities of the financial series, has a positive efect when obtaining an approximation for the average value of the series. In the same way and as in applying the forecast to one step, there is a clear advantage in having the possibility of modeling the variance through time, which is at the same time a more realistic representation of the behavior of the TRM.
(
The TRM is one of the most important economic items for the Colombian economy. This financial series is influenced by various factors originated by national and international political and economic environment, adding stochastic behaviors to its structure and high volatility clusters, making its forecast quite dificult to carry out.
In this paper the TRM series was studied over a longer period of time (from 2013-01 to 2020-07) compared with previous works, with the aim to propose a more robust time series model. For this, an iterative methodology was proposed for the fitting, selection and forecast of the most adequate models from three classes of time series: ARMA, ARMA-GARCH and Markov Switching (MS). The latter two with the property that they can model the non-stationary behavior of the TRM variance.
A MS model with 3 regimes and (0) processes achieved the best fit over the ARMA(
In conclusion, the possibility to characterize several realities through the regimes of MS models gives the opportunity of obtaining a better description of the TRM series in a longer time window, and in the GARCH model to describe the stochastic nature of volatility. For future works it seems necessary to evaluate an hybrid GARCH-MS model that could forecast more accurately the abrupt changes in mean and variance of the TRM series.